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Automating motion trajectory Of crane-lifted loads
Automation in Construction 45 (2014) 178–186
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Automation in Construction journal homepage: www.elsevier.com/locate/autcon
Automating motion trajectory Of crane-lifted loads Jacek Olearczyk a,1, Ahmed Bouferguène b,⁎, Mohamed Al-Hussein c,2, Ulrich (Rick) Hermann d,3 a
Dept. of Civil and Environmental Engineering, University of Alberta, T6G 2W2, Canada Campus Saint Jean, University of Alberta, 8406-91 Street, Edmonton, Alberta, Canada c Dept. of Civil and Environmental Engineering, University of Alberta, T6G 2W2, Canada d PCL Industrial Management Inc., 5404-99 Street, T6E 3P4 Edmonton, Alberta, Canada b
a r t i c l e
i n f o
Article history: Received 4 February 2013 Received in revised form 2 June 2014 Accepted 8 June 2014 Available online 2 July 2014 Keywords: Mobile crane lifts Object trajectory Polar coordinates Mathematical algorithm
a b s t r a c t Crane Lifting Path Planning (CLPP) is an important task, especially in congested construction sites. This activity becomes extremely complex since, as the project advances, new (and sometimes unpredicted) constraints may occur. These constraints force lift engineers to explore the possibility of alternative paths for the objects still to be moved. Under severe time constraints, manual analysis of such paths is practically impossible, which makes it necessary to rely on computer implementations in order to avoid project delays. From a mathematical standpoint, if the crane payload trajectory is to be defined analytically, the polar coordinate system,(r, θ), is naturally the most suitable. As a result, this paper proposes an algorithm for CLPP in which the path is represented as a piece-wise continuous function where each portion is defined either by constant radius (rotation of the load) or a constant angle (translation of the load). In other words, rather than forcing a crane to adapt to unnatural rectilinear trajectories as obtained by traditional path searching procedures, it is the trajectory that is adapted to the crane motion. In fact, even though (mathematically) a precise balance between the rates of rotation and translation of the jib (or boom) will make the payload follow any continuous path (regardless of its complexity), the coupling between the rotations and translations increases the difficulty of the lifting activity. However, a crane lifting path in which these motions are uncoupled will lead to less stringent requirements in terms of controlling the balance between rotations and translations. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Using computer technology to design construction equipment has brought changes to equipment functionality, productivity, and the construction industry. Mobile cranes, in particular, have increased in size, capacity, manoeuvrability, and versatility. The significance of misusing or improperly planning a crane lift can be severe. The preparation and planning of heavy lifts (usually 85% of crane lift capacity or more) can be complex and involve challenges during lift analysis. Fast-tracked construction projects involve frequent changes to the construction plan, thereby requiring a number of what-if scenarios and modifications to the lift plan. Due to these factors, lift engineers rely on complex computer algorithms to make informed decisions in which there is no room for guess work. Virtual reality, combined with an interactive planning environment that explores human potential, is ⁎ Corresponding author. Tel.: +1 780 465 8719. E-mail addresses: [email protected] (J. Olearczyk), [email protected] (A. Bouferguène), [email protected] (M. Al-Hussein), [email protected] (U.(R.) Hermann). 1 Tel.: +1 780 913 3599. 2 Tel.: +1 780 492 0599; fax: +1 780 492 0249. 3 Tel.: +1 780 733 5514; fax: +1 450 733 5779.
http://dx.doi.org/10.1016/j.autcon.2014.06.001 0926-5805/© 2014 Elsevier B.V. All rights reserved.
an efficient alternative for spatial integration. In a well-known article introducing spatial integration in construction [2], has presented the historical advancement and objectives of this technology. In other studies, researchers have borrowed concepts from the field of robotics, e.g., configuration space (C-Space), which has opened a new approach for generating crane lifting paths [7]. Generating the shortest collision-free paths for crane lifting operations is an important challenge for the (heavy) construction industry, especially since it moved toward modularization as one of its major project delivery paradigms. As a result, there is a growing need for mathematical models and technologies that can help managers to precisely and easily plan crane operations, regardless of the level of congestion encountered on site. From the point of view of technological innovation, the introduction of sensing devices to improve safety and efficiency of crane operations has opened new research opportunities, especially in the context of sites with multiple cranes where coordination is paramount. In this context, Lee et al. [5] have developed a laser-based lifting-path tracking system for a robotic tower crane system. However, although this technology is expected to improve productivity by as much as 50%, further investigations are needed in order to determine its robustness when subject to the harsh conditions of construction sites, e.g., dust, vibrations and rainfall. In a subsequent
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contribution, the sensing technology introduced by these authors was integrated within a Building Information Model (BIM) in order to extend its application to the case where a tower crane is operated in the presence of blind spots, i.e., blind lifts [4]. In terms of mathematical modeling and algorithms, the importance of Crane Lifting Path Planning (CLPP) in the construction industry has drawn the attention of many researchers, who have proposed a variety of procedures allowing lifted objects to be moved around obstacles. These algorithms, known collectively as motion planning, path finding, or path searching, have been well developed in the field of computer science and can be viewed as graph traversal methods in which some form of optimality is sought. Among the most widely used procedures for optimal path finding are: (i) Dijkstra's [10,12]; (ii) the A* (A star) [10,9]; (iii) evolutionary algorithms including Genetic algorithm optimization [10]; (iv) the Configuration Space (C-space) [7,3]; and (v) Rapidly-exploring Random Trees (RRT), not to mention the numerous variations that were built on top of the above listed algorithms. In the specific context of CLPP, it is worth noting that Soltani et al. [10] have evaluated the performance of three of the above algorithms, namely Dijkstra's, A* and GA. They concluded that: (i) Dijkstra's method can find an optimal path, but that its efficiency deteriorates as the complexity of the problem at hand increases; (ii) the A* approach, being essentially an optimal form of Dijkstra's, is relatively more efficient than its predecessor but still becomes impractical for large-scale problems; and (iii) GA based path searching is naturally probabilistic, which means that its solutions may be less accurate than the first two, but that it has the ability to remain efficient even in the case of problems with large dimensionality. More recently, Zheng and Hammad [13] have developed a remarkable variation of the RRT algorithm, referred to as RRT-Con-Con-Mod, which allows paths to be generated dynamically as a means to improve efficiency and safety, especially when unexpected changes are encountered on construction sites. A smoothing procedure is applied to the paths generated by the RRT-Con-Con-Mod algorithm in order to reduce their roughness, which allows unnecessary movements to be avoided. Historically, path finding methods have been developed primarily in the context of robotics and have been geared toward determining the shortest obstacle-free path between two points. As a result, these paths are often represented as polylines, i.e., piece-wise continuous linear functions, since straight lines are very simple yet well adapted trajectories for a moving robot. However, while such a topology is coherent with respect to the way robots move, it is unnatural for a crane payload, since at a fixed elevation its trajectory regardless of its complexity is generated by rotating and translating selected components of the crane. In other words, a rotation and translation can be viewed as a basis that can be used to describe a crane lifting path in ! ! the same way as i and j are used to describe any vector in a 2-D plane. Interestingly, Kang & Miranda [3], who developed a procedure based on the C-space, raised the issue of polyline trajectories and devised a procedure to smooth the piece-wise linear path in order to obtain more realistic trajectories. However, although this smoothing process yields a curvilinear path, the crane operator may still require a great deal of control in order to balance the rotation and translation speeds in order for the payload to follow this path. The main objective of this contribution is to provide a CLPP alternative that contributes to solving the problem of rectilinear crane lifting paths. Indeed, rather than using standard robot path planning algorithms and then adapting (to some extent) the resulting trajectory to the specifics of a crane, the proposed algorithm seeks a trajectory that can be expressed as a sequence of simple rotations and translations in which smoothness— i.e., the number of rotation-to-translation transitions—is minimized. This approach is fundamentally different from previous work. In the section which follows, this paper presents the two basic building blocks underlying the algorithms described in this work: (i) the decomposition of a crane lifting path in terms of the basic motions of a crane; and (ii) a brief review on developing and merging polygonal envelopes around obstacles. In Section 3, we review some fundamental
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properties of the A* algorithm, which has been selected for this research because of its simplicity. Issues related to the topology of the paths obtained using robot path planning procedures are also discussed in the context of CLPP. In Section 3.1 an initial (and less efficient) algorithm using rotations and translations is proposed. It starts by building around obstacle envelopes shaped in the form of circular sections whose centers are the crane locations. Once overlapping areas are merged, a path following the contours of these circular sections is constructed by means of a binary tree. Although useful on sites with little congestion, this algorithm may fail to find a lifting path when the congestion increases. This is due to the fact that merging the circular sections, which often are much larger than the obstacles, tends to unnecessarily exclude large portions of the construction site. The second algorithm, described in Section 3.2, is developed to solve this very issue. In the context of this CLPP procedure, the envelopes around the obstacles are constructed in such a way as to exclude the least area of the construction site. A circular grid centered on the crane location is overlaid on the construction site and then mapped onto a matrix where cells containing obstacles receive an infinite weight. This matrix is used for path searching using the A* algorithm. Finally, we provide a case study in which this algorithm was used to assess its practical usefulness.
2. Methodology Over the past few decades, modular construction has emerged as the paradigm of choice for heavy industrial construction, allowing practitioners to reduce the uncertainty related to harsh and unpredictable weather conditions. Module fabrication, which relocates construction activities to controlled environments (i.e., fabrication plants), has become increasingly efficient as optimization techniques such as lean manufacturing are utilized. As expertise in modularization has improved, construction professionals have sought to increasingly integrate modular construction technologies into their projects. Meanwhile, the fabricated modules have become heavier and more voluminous, thus requiring high capacity cranes for on-site assembly. Consequently, minute planning of crane lifts has become a crucial activity for managers in order to ensure safe, timely, and economical project delivery. This is particularly important on congested sites where the risk of collisions between the payload and the surrounding obstacles is high. In essence, for a given load which is to be moved, at a fixed elevation, from its Pick Point (PP) to its Set Point (SP), the optimal path (in the absence of obstacles) can be described by Fig. 1, in which the object's trajectory is represented by a dotted line. Under the fixed elevation assumption, the path of any object can be represented by a sequence of rotations and inward or outward translation. For instance, the path ℘(PP → SP) from the pick point PP to the
SP
PP
Fig. 1. Crane lifting path for a load picked at PP and set at SP in the absence of obstacles. The path is represented as a dotted line.
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(a)
(b) 1
Obsstaclle
Obbstaccle
2
3
Fig. 2. Modifying a simple trajectory to avoid an obstacle: (a) collision detection and (b) collision-free path.
set point SP shown in Fig. 1 can be formalized according to Eq. (1), ℘ðPP→SPÞ ¼ ℜ ðR1 ; α Þ⊕ℑ ðR2 −R1 Þ
ð1Þ
In which ℜ represents the rotation of angle α at a constant radius R1 (with respect to the position of the crane), whereas ℑ denotes the outward translation whose vector magnitude is defined as R2 − R1. The outward translation is characterized by R1 b R2 and the inward translation by R1 N R2. As for the symbol ⊕, it is used to denote the composition of the translation ℑ(R2 − R1) and the rotation ℜ(R1, α). In practice, for any lift performed at a fixed elevation, two possibilities may be encountered: (i) No obstacle interferes with the object in which case the path of the lifted object is straightforward, see Fig. 1; (ii) One or more obstacles lie on the path of the object (or the boom). In such a case, a collision resolution procedure must be applied in order to obtain a collision-free path. In what follows, we describe an algorithm which modifies the simple payload trajectory shown in Fig. 1 in order to avoid obstacles while satisfying user-defined clearance requirements. This aspect is illustrated in Fig. 2(b) where it can be seen that even though the path becomes more complex than that in Fig. 1, its formal representation is still expressed in terms of the crane basic degrees of motions (at a fixed elevation), i.e., rotations and inward/ outward translations. According to (Fig. 2), avoiding an obstacle at a given elevation can be viewed as a process of modifying a simple trajectory, see Fig. 2a, so to obtain an obstacle-free path, shown in Fig. 2b, which can be formalized satisfying Eq. (2), ℘ðPP→SPÞ ¼ ℜ ðR1 ; α 1 Þ⊕ℑ ðR2 −R1 Þ⊕ℜðR2 ; α 2 Þ⊕ℑ ðR3 −R2 Þ⊕ℜðR3 ; α 3 Þ ð2Þ
(a)
From a high level conceptual point of view, the two algorithms proposed in this work for CLPP proceed according to the following steps: (i) around each obstacle, build an envelope representing an exclusion area which, for safety reasons, should not be entered by the payload to ensure sufficient clearance between the object and surrounding obstacles; (ii) In the case where obstacles' envelopes overlap, merge these envelopes into a single exclusion area; (iii) apply a path searching algorithm, starting at the pick point and ending at the setpoint, that builds an obstacle-free path (optimal with respect to a predefined metric). Here it is worth noting that the proposed algorithms differ in the way they handle step (iii).
2.1. Obstacle envelopes According to the high level description provided in the previous section, the first step towards determining the trajectory of a lifted object consists of constructing an envelope around each obstacle which exists within the crane's operation area. For safety reasons, any lifted object must remain outside such envelopes during its travel from the pick point to its final resting set point. The algorithm allowing an envelope to be constructed around a polygonal obstacle proceeds in three steps: (i) an initialization phase in which the first edge of the envelope is obtained, followed by (ii) a propagation step which generates the remaining edges and finally (iii) a merging phase in which overlapping envelopes are merged into a single area. The first two steps are illustrated in Fig. 3. Although the initialization of the algorithm is rather straightforward, the following implementation details are provided to enhance the readability of this contribution:
(b)
Fig. 3. Obstacle envelope generation. (a) Initialization and (b) propagation.
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Valid envelop segments Invalid segments
Fig. 4. Rules for the construction of edges of the envelope in the propagation phase.
1. Find two consecutive points A (xA, yA) and B (xB, yB) such that yA and yB are, respectively, the smallest and second smallest ordinates, that is yA = min({yk}k = 1,2,⋯,n) andyB = min({yk}k = 1,2,⋯,n − {yA}). 2. The first edge of the envelope (shown in Fig. 3) is a segment [A ' B '] such that: 8 0 0 < A B ==½AB 0 0 : d A B ; ½AB ¼ User specified clearance yA0 byA and yB0 byB
ð2Þ
In which the notation d([A ' B '], [AB]) represents the distance between segments [A ' B '] and [AB]. As for the propagation phase, the application of the distance requirement (given in Eq. (2)) yields for each subsequent edge two segments of which only one is valid since it belongs to the envelope, see Fig. 4. The selection algorithm rules are described in Fig. 4. According to Fig. 4, the valid segment is chosen according to the following rule: a. For α b π/2, select the segment for which d1 N d2 and b. for α N π/2, select the segment for which d1 b d2. Where d1 and d2 are the distances from the end point of the current (valid) edge of the envelope to the closest end points of the next edges, see Fig. 4. At this stage, the obstacle envelope is represented by a sequence of unconnected edges which needs to be connected in order to obtain a closed polygonal shape as shown in Fig. 5. For simplicity, the connections are obtained by finding the intersection of consecutive edges. It is worth mentioning that the obtained envelope is continuous but is clearly not smooth (not differentiable) at the points of connection. For the purpose of this contribution, this does not create any limitation on the subsequent algorithms but if needed smoothness can be achieved by connecting the segments by ad-hoc curvilinear arcs.
need to adapt to changing layouts, e.g., bridges becoming unusable. Because the A* algorithm combines efficiency and simplicity [8] it is among the most widely used path finding procedures in the gaming industry. The A* was built on Dijkstra's graph traversal approach but departs from its predecessor since the search from the current node (n) onward is directed by means of a cost function calculated satisfying Eq. (3), f ðnÞ ¼ g ðnÞ þ hðnÞ
ð3Þ
In which g(n) is the total cost required to get from the starting point to the current location and h(n) is a heuristic function used to estimate the cost from the current node to the final location, i.e., the target. For instance, given a layout, a starting node and a target, see Fig. 7, the A* algorithms returns the path represented in red. However, in the context of CLPP, even though the A* family of algorithms provides valid paths, their geometry does not reflect the motions allowed for a crane which as mentioned in the previous section has only two degrees of freedom at a fixed elevation: rotation and translation. As a result, for CLPPs the A* (or equivalent algorithms) results can mostly be characterized as proofs of existence of paths since their rectilinear shape does not necessarily describe what happens on the field. To illustrate this issue, let us assume that a payload is moved on a construction site free of obstacles. In this case, the application of any path planning algorithm would return a straight line which conveys
2.2. Merging individual envelopes As the construction process advances, objects placed in their final positions may interfere with the trajectories of those waiting to be lifted. As a result, envelopes need to be re-generated after each lift, and since some of these may overlap, it may be necessary to merge them into a single larger and more complicated envelope. For the purpose of this contribution, the Weiler–Atherton algorithm [11,1] was used. The result of this procedure is illustrated in Fig. 6. 3. Path searching Path searching is a well-known problem in artificial intelligence, which over the past few decades has led to the development of several algorithms [7,6] especially in connection with the emergence of computer games. Indeed, the increasing complexity of modern (strategic) games required sophisticated path finding procedures which not only need to be fast in order to ensure user-friendliness but most importantly
Fig. 5. Obstacle clearance envelope obtained by connecting the edges of the propagation phase described in Figs. 3 and 4.
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Fig. 6. Illustration of the Weiler–Atherton polygon union algorithm.
the idea of a shortest distance; however, to mimic this trajectory the crane operator would need a great deal of control on the variation of the radius of the payload. Indeed, if we assume the crane base is located at the origin (xc = 0, yc = 0), and the linear path is described by an equation of the form y = mx + b, see Fig. 8, the radius of the payload, R(θ), which varies as a function of the rotation angle (for the load to follow the line) is given by Eq. (4), 8 > < θPP ¼ b with RðθÞ ¼ > sinðθÞ−m cosðθÞ : θSP ¼
π 2 π −1 tan ðySP =xSP Þ þ ½1− sgnðxSP Þ 2 −1
tan
ðyPP =xPP Þ þ ½1− sgnðxPP Þ
ð4Þ In which θPP ≤ θ ≤ θSP is the rotation angle of the jib (or boom), (x PP, y PP) and (xSP, ySP) are respectively the coordinates of the pick point (PP) and the set point (SP) whereas the function sgn(x) represents the sign function which returns 1 if x ≥ 0 and −1 otherwise. In Eq. (4), because the payload trajectory is defined analytically, we are able to describe explicitly the coupling between the radius (distance of the load from the crane base) and the rotation angle. Even though it is challenging for (human) crane operators to precisely balance the speeds of rotation and translation which would allow the load to follow complicated trajectories, analytical forms like that in Eq. (4) may be handy for robot driven cranes. However, in the case where the trajectory is obtained by means of ad-hoc computational procedures the coupling between
Fig. 7. Finding a path connecting a source and a destination using the A* algorithm.
the radius and the rotation angle can only be done by discretization since the trajectory is not defined explicitly. In this case, the crane operator may decide to perform a sequence of rotations–translations as shown in Fig. 9 (in the case of a linear lifting path) where three different scenarios are proposed based on the length of the corresponding path. This approach which can be viewed as a discretization of the continuous motion described in Eq. (4) is time consuming since it requires many rotation-to-translation transitions. Consequently, the intrinsic degrees of freedom of the crane need to be integrated as constraints within the procedures used for payload path generation in order to avoid un-natural scenarios. In view of the difficulty in using linear paths as determined by traditional path planning algorithms, it is instructive to explore the challenges related to determining a path which uses only the basic crane motions. At this juncture, it is important to note that even though this procedure will not necessarily lead to the shortest path, it is practically more realistic. In fact, rather than determining a path and force the crane to follow it, we propose to determine a path that is adapted to the crane's constrained motion, rotation, and translation. 3.1. First algorithm: motion along the outskirt of (merged) obstacle envelops The starting point of the procedure developed in this contribution is a simple trajectory which consists of a single rotation and an inward or outward translation connecting the pick to the set point for a given object, Fig. 1. This trajectory is referred to as the Crane Simple Trajectory
Fig. 8. Moving a payload along a linear path by adjusting in a continuous way its radius as a function of θ.
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Fig. 9. Sequence of crane rotations/translations to approximate a linear path obtained by path planning algorithms in the absence of obstacles. The rotation/translation paths are ranked based on their lengths.
(CST). In the case where an obstacle interferes with the CST, an inward or outward translation is applied to the CST to avoid the obstacles (assuming a fixed elevation). When the lifted object is translated, the inward/outward movement must be within the minimum and maximum radii allowed for that weight, see Fig. 2. From an algorithmic point of view, the search for a path connecting the pick point of an object to its final resting position relies on a binary tree the nodes of with nodes are created every time an obstacle is encountered. The branches of each node allow two options: inward or outward translation immediately followed by a rotation towards the set point. In essence this algorithm moves the object along the outskirts of the obstacles lying on its path which are identified by circular sections, as shown in Fig. 10. Overlapping sections will be merged to produce a single envelope. To allow the practitioner to select an optimal path from those generated by the new algorithm, each rotation inward/outward translation is assumed to carry a penalty which is likely to be project dependent. For instance, 1. A transition from a translation to a rotation or vice versa has an intrinsic penalty which quantifies the unproductive time during which the crane operator transitions from one motion to another. 2. An object which is translated outwardly to avoid an obstacle leads to an increase in the moment due to its weight, which in turn, increases the likelihood of crane failure due to a negative reaction exerted on its base. This increased risk is particularly true in the presence of additional forces such as wind.
+
Building on the previous step, in which each obstacle was surrounded by a sector, see Fig. 10, the most straightforward algorithm allowing an object to be moved by a crane from its pick-point to its final destination consists in following the contours of the obstacles' envelopes. Incidentally, such envelopes are built using arcs and forward/backward translation, hence taking into account the specifics of the crane motions. The possible paths are generated using a binary tree to which penalties (or weights) are assigned which allows the most optimal to be selected. This concept is illustrated in Fig. 11, in which the possible paths are shown. Although the paths determined in Fig. 11 are likely not optimal since they contain several rotation-to-translation transitions, these were generated as in such a way as to be expressed in terms of the fundamental motions of a crane: rotations and translations both of which are uncoupled since each motion occurs while the other is constant. Of course, since every transition from rotation to translation (and vice versa) causes delay, the above described algorithm returns the path which contains the least amount of such transitions. 3.2. Second algorithm: motion along the outskirt of individual (unmerged) obstacle envelopes While the above described algorithm has the advantage of simplicity, it has a major drawback: merging envelopes may dramatically decrease the area in which an object can be moved hence increasing (falsely) the likelihood of impossible lifts. In the following section, we
+
Fig. 10. Circular sections around obstacles. The object is moving along segments of the envelopes which are shown in red. The crane location is identified by the white cross.
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Fig. 11. Pick point to final destination paths for an object moving along the outskirt of the obstacles. When superimposed, the individual paths on the left give the red contour on the right.
describe a more general, yet more complex, approach allowing one to obtain an optimal path for a given object. This algorithm proceeds in four steps: 1. Apply a rotation and a translation to the site so as to put the pick location on the X axis and the crane location at the origin. 2. From the location of the crane (xc = 0, yc = 0), draw rays connecting the crane to each vertex of the obstacles. 3. For each vertex (including the pick and set points), a circular arc is drawn starting from the X axis (containing the pick point) to the ray which contains the set position. 4. The intersections of the rays with the arcs are compiled into a matrix, in which the columns correspond to the angles (smallest to largest) and the lines represent the radiuses (smallest to largest). Nodes which fall within an obstacle are given an infinite weight. Given the matrix (which can also be viewed as a graph) of the possible nodes, the next step consists of applying any known algorithm (including the A* algorithm) to find the shortest path between the node representing the pick point to the final destination. The matrix corresponding to the case depicted in Fig. 11, which was made more complicated by the addition of two circles representing the minimum and maximum reaches of the crane, is given in Fig. 12. In this table, “x”
Fig. 12. Matrix representation of the ray–arc intersections. Red symbols denote forbidden locations. Blue indicates the pick and set points of the object.
denotes the nodes that fall in forbidden areas (infinite weight), whereas as “1” indicates allowed positions. Although the matrix in Fig. 12 (which essentially is a graph) provides a number of possible routes linking the pick to the set point, in practice one may choose the path with the least number of rotations to inward/ outward translation transitions. An example of such paths is depicted in Fig. 13 in which the most optimal route contains three rotation-totranslation transitions. Note that in the layout described in Figs. 12 and 13, the payload was purposely forced to be moved in the upper half of the site, i.e., the most congested area, to illustrate the flexibility of the procedure described in this work. In practice, however, the algorithm described in this paper above allows the boom to rotate in both directions: clockwise (lower half) and counter clockwise (upper half) and returns the smoothest path.
4. Case study The algorithm developed in this work was applied to a case study in the field of heavy industrial construction site where heavy modules needed to be assembled into a complex structure to be used in the oil industry in Northern Alberta. In this context, a typical project may require assembling 50 to more than 300 modules. As the project progresses, the construction site becomes increasingly congested
Fig. 13. Selected paths constructed from the matrix mapping of the ray–arc intersections. The inner and outermost circles represent the minimum and maximum reaches of the crane.
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SP
PP Fig. 14. A plane and isometric view of a module to be lifted by a crane from a Pick Point (PP) and installed at the end of a corridor like structure identified as the Set Point (SP).
hence requiring minute planning of the remaining lifts in order to ensure a level of productivity allowing scheduling and safety constraints to be met. This is particularly important for the Northern part of Alberta where the harsh winter conditions (temperature and wind factor) can be challenging. To illustrate the usefulness of the algorithm developed in this work, we provide a case where a module picked from a location (PP) is required to be installed at the end of a corridor-like structure at a location SP, see Fig. 14. In the case portrayed in Fig. 14, a DEMAG CC2800 crawler crane whose maximum capacity is 660 t at a maximum boom length of approximately 84 m (275 ft) was used. The crane was equipped with a superstructure to ensure that all modules could be assembled safely. The module of interest shown in Fig. 14 weighed approximately 65 metric tons for a size of 12 × 6 × 7.21 m. Referring to Fig. 14, the module of interest, picked from a trailer, Fig. 14(b), is required to be installed at the other end of a two-part structure which exhibits a corridor like area, Fig. 14(a). Although, the module could be lifted above existing structures, the preference of our industrial partner is to lift as close to the ground as possible before considering lifting at higher elevations since the former scenario provides a better control of the load especially in the presence of strong wind. Given the pick and set positions of the module shown in Fig. 14, the application of A* and the C-space algorithms returned a linear path which requires the load to be moved in the corridor since both of these procedures target the shortest distance as their objective function, Fig. 15. However, although valid, these paths were found not convenient from a practical point of view since they require a great deal of control in order to move the module of interest without collisions with the surrounding obstacles. Conversely, the procedure developed in this work was able to determine an alternative path since the metric
underlying the validity of a path is its smoothness measured in terms of the number of transitions from rotation to translation (and vice versa). As a result, given the fact that the reach of the crane used for this operation allowed for the module of interest to be moved at a larger radius, the path from the pick to the set point in the exact same form as Eq. (1), that is one outward translation, one counter clockwise rotation followed by an inward translation. This type of paths which are constrained to use the basic motions of a crane turned out to be easier to follow by crane operators. 5. Conclusion Cranes are among the most important equipment used in today's construction industry, especially in the context of modular construction, where heavy modules built off-site need to be assembled on site according to time-constrained schedules. This is particularly true in the case of Northern Alberta which is characterized by harsh winter conditions which can reduce the window for outdoor work. As a result, cranes, which over the years have increased in size and capacity to accommodate the requirements of modular technology, have become a critical resource on construction sites. The utilization of these cranes, though, requires minute planning in order to reduce the risks of failures and delays. In this respect, optimizing this resource has become an important task that can no longer be overlooked by project planners and practitioners. The main thrust of this contribution is thus the development of an alternative CLPP approach which has the advantage of generating paths that are constrained to be represented in terms of the basic motions of cranes, namely, rotations and translations. In fact, the main issue in using traditional path planning algorithms is the shape of the resulting paths, which are expressed as a polyline. From a
Fig. 15. Module path as obtained by the A* and C-space algorithms which optimize the shortest traveling distance.
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mathematical point of view, this topology which does not take into account the specifics of the crane motions ends up coupling the radius (distance of the payload from the crane) to the rotation angle in order to mimic trajectories which are not easily expressible in terms of the intrinsic motions of a crane. In practice this could be carried out by precisely controlling and balancing the rates of rotation and translation which can be very challenging especially for inexperienced crane operators working in congested sites. As a result, rather than applying path planning techniques that ultimately lead to linear paths, this work proposes to explore the possibility of generating paths that can easily be expressed as a sequence of rotations and translations in accordance with the basic motions that can be performed by a crane jib (or boom). The proposed algorithm aims at complementing existing CLPP procedures in order to improve the efficiency of crane operations. Acknowledgments The research presented in this paper has been financially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), project number CRDPJ 380518-08. References [1] P. Atherton, K. Weiler, D. Greenberg, Polygon shadow generation, Comput. Graph. 12 (3) (1978) 275–281.
[2] L. Bernold, Spatial integration in construction, ASCE J. Constr. Eng. Manage. 128 (5) (2002) 400–408. [3] S. Kang, E. Miranda, Planning and visualization for automated robotic crane erection processes in construction, Autom. Constr. 15 (2006) 398–414. [4] G. Lee, J. Cho, S. Ham, T. Lee, G. Lee, S. Yun, H. Yang, A BIM- and sensor-based tower crane navigation system for blind lifts, Autom. Constr. 26 (2012) 1–10. [5] G. Lee, H. Kim, C. Lee, S. Ham, S. Yun, H. Cho, B.K. Kim, G.T. Kim, K. Kim, A lasertechnology-based lifting-path tracking system for a robotic tower crane, Autom. Constr. 18 (2009) 865–874. [6] T. Miyoshi, S. Kawakami, K. Terashima, Path planning and obstacle avoidance considering rotatory motion of load for overhead cranes, J. Mech. Syst. Transp. Logist. 1 (1) (2008) 134–145. [7] R. Raghunatha, K. Varghese, Automated path planning for mobile crane lifts, Comput. Aided Civ. Infrastruct. Eng. 17 (2002) 439–448. [8] S.J. Russell, P. Norvig, Artificial Intelligence: A Modern Approach, Prentice Hall, Upper Saddle River, NJ, USA, 2003. [9] P. Sivakumar, K. Varghese, N.R. Babu, Automated path planning of cooperative crane lifts using heuristic search, ASCE J. Comput. Civ. Eng. 17 (3) (2003) 197–207. [10] A.R. Soltani, H. Tawfik, J.Y. Goulermas, T. Fernando, Path planning in construction sites: performance evaluation of the Dijkstra, A*, and GA search algorithms, Adv. Eng. Inform. 16 (4) (2002) 291–303. [11] K. Weiler, P. Atherton, Hidden surface removal using polygon area sorting, Proceedings of SIGGRAPH '77, Computer Graphics', 1977, pp. 214–222. [12] S. Wengui, L. Li, Z. Xiaorong, L. Xiaoping, Application of Optimized Dijkstra Algorithm in Storage/Retrieval Routes Scheduling of a Stacker Crane, 2009. [13] C. Zhang, A. Hammad, Improving lifting motion planning and re-planning of crane with consideration for safety and efficiency, J. Adv. Eng. Inf. 26 (2) (2012) 396–410.
Contents lists available at ScienceDirect
Automation in Construction journal homepage: www.elsevier.com/locate/autcon
Automating motion trajectory Of crane-lifted loads Jacek Olearczyk a,1, Ahmed Bouferguène b,⁎, Mohamed Al-Hussein c,2, Ulrich (Rick) Hermann d,3 a
Dept. of Civil and Environmental Engineering, University of Alberta, T6G 2W2, Canada Campus Saint Jean, University of Alberta, 8406-91 Street, Edmonton, Alberta, Canada c Dept. of Civil and Environmental Engineering, University of Alberta, T6G 2W2, Canada d PCL Industrial Management Inc., 5404-99 Street, T6E 3P4 Edmonton, Alberta, Canada b
a r t i c l e
i n f o
Article history: Received 4 February 2013 Received in revised form 2 June 2014 Accepted 8 June 2014 Available online 2 July 2014 Keywords: Mobile crane lifts Object trajectory Polar coordinates Mathematical algorithm
a b s t r a c t Crane Lifting Path Planning (CLPP) is an important task, especially in congested construction sites. This activity becomes extremely complex since, as the project advances, new (and sometimes unpredicted) constraints may occur. These constraints force lift engineers to explore the possibility of alternative paths for the objects still to be moved. Under severe time constraints, manual analysis of such paths is practically impossible, which makes it necessary to rely on computer implementations in order to avoid project delays. From a mathematical standpoint, if the crane payload trajectory is to be defined analytically, the polar coordinate system,(r, θ), is naturally the most suitable. As a result, this paper proposes an algorithm for CLPP in which the path is represented as a piece-wise continuous function where each portion is defined either by constant radius (rotation of the load) or a constant angle (translation of the load). In other words, rather than forcing a crane to adapt to unnatural rectilinear trajectories as obtained by traditional path searching procedures, it is the trajectory that is adapted to the crane motion. In fact, even though (mathematically) a precise balance between the rates of rotation and translation of the jib (or boom) will make the payload follow any continuous path (regardless of its complexity), the coupling between the rotations and translations increases the difficulty of the lifting activity. However, a crane lifting path in which these motions are uncoupled will lead to less stringent requirements in terms of controlling the balance between rotations and translations. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Using computer technology to design construction equipment has brought changes to equipment functionality, productivity, and the construction industry. Mobile cranes, in particular, have increased in size, capacity, manoeuvrability, and versatility. The significance of misusing or improperly planning a crane lift can be severe. The preparation and planning of heavy lifts (usually 85% of crane lift capacity or more) can be complex and involve challenges during lift analysis. Fast-tracked construction projects involve frequent changes to the construction plan, thereby requiring a number of what-if scenarios and modifications to the lift plan. Due to these factors, lift engineers rely on complex computer algorithms to make informed decisions in which there is no room for guess work. Virtual reality, combined with an interactive planning environment that explores human potential, is ⁎ Corresponding author. Tel.: +1 780 465 8719. E-mail addresses: [email protected] (J. Olearczyk), [email protected] (A. Bouferguène), [email protected] (M. Al-Hussein), [email protected] (U.(R.) Hermann). 1 Tel.: +1 780 913 3599. 2 Tel.: +1 780 492 0599; fax: +1 780 492 0249. 3 Tel.: +1 780 733 5514; fax: +1 450 733 5779.
http://dx.doi.org/10.1016/j.autcon.2014.06.001 0926-5805/© 2014 Elsevier B.V. All rights reserved.
an efficient alternative for spatial integration. In a well-known article introducing spatial integration in construction [2], has presented the historical advancement and objectives of this technology. In other studies, researchers have borrowed concepts from the field of robotics, e.g., configuration space (C-Space), which has opened a new approach for generating crane lifting paths [7]. Generating the shortest collision-free paths for crane lifting operations is an important challenge for the (heavy) construction industry, especially since it moved toward modularization as one of its major project delivery paradigms. As a result, there is a growing need for mathematical models and technologies that can help managers to precisely and easily plan crane operations, regardless of the level of congestion encountered on site. From the point of view of technological innovation, the introduction of sensing devices to improve safety and efficiency of crane operations has opened new research opportunities, especially in the context of sites with multiple cranes where coordination is paramount. In this context, Lee et al. [5] have developed a laser-based lifting-path tracking system for a robotic tower crane system. However, although this technology is expected to improve productivity by as much as 50%, further investigations are needed in order to determine its robustness when subject to the harsh conditions of construction sites, e.g., dust, vibrations and rainfall. In a subsequent
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contribution, the sensing technology introduced by these authors was integrated within a Building Information Model (BIM) in order to extend its application to the case where a tower crane is operated in the presence of blind spots, i.e., blind lifts [4]. In terms of mathematical modeling and algorithms, the importance of Crane Lifting Path Planning (CLPP) in the construction industry has drawn the attention of many researchers, who have proposed a variety of procedures allowing lifted objects to be moved around obstacles. These algorithms, known collectively as motion planning, path finding, or path searching, have been well developed in the field of computer science and can be viewed as graph traversal methods in which some form of optimality is sought. Among the most widely used procedures for optimal path finding are: (i) Dijkstra's [10,12]; (ii) the A* (A star) [10,9]; (iii) evolutionary algorithms including Genetic algorithm optimization [10]; (iv) the Configuration Space (C-space) [7,3]; and (v) Rapidly-exploring Random Trees (RRT), not to mention the numerous variations that were built on top of the above listed algorithms. In the specific context of CLPP, it is worth noting that Soltani et al. [10] have evaluated the performance of three of the above algorithms, namely Dijkstra's, A* and GA. They concluded that: (i) Dijkstra's method can find an optimal path, but that its efficiency deteriorates as the complexity of the problem at hand increases; (ii) the A* approach, being essentially an optimal form of Dijkstra's, is relatively more efficient than its predecessor but still becomes impractical for large-scale problems; and (iii) GA based path searching is naturally probabilistic, which means that its solutions may be less accurate than the first two, but that it has the ability to remain efficient even in the case of problems with large dimensionality. More recently, Zheng and Hammad [13] have developed a remarkable variation of the RRT algorithm, referred to as RRT-Con-Con-Mod, which allows paths to be generated dynamically as a means to improve efficiency and safety, especially when unexpected changes are encountered on construction sites. A smoothing procedure is applied to the paths generated by the RRT-Con-Con-Mod algorithm in order to reduce their roughness, which allows unnecessary movements to be avoided. Historically, path finding methods have been developed primarily in the context of robotics and have been geared toward determining the shortest obstacle-free path between two points. As a result, these paths are often represented as polylines, i.e., piece-wise continuous linear functions, since straight lines are very simple yet well adapted trajectories for a moving robot. However, while such a topology is coherent with respect to the way robots move, it is unnatural for a crane payload, since at a fixed elevation its trajectory regardless of its complexity is generated by rotating and translating selected components of the crane. In other words, a rotation and translation can be viewed as a basis that can be used to describe a crane lifting path in ! ! the same way as i and j are used to describe any vector in a 2-D plane. Interestingly, Kang & Miranda [3], who developed a procedure based on the C-space, raised the issue of polyline trajectories and devised a procedure to smooth the piece-wise linear path in order to obtain more realistic trajectories. However, although this smoothing process yields a curvilinear path, the crane operator may still require a great deal of control in order to balance the rotation and translation speeds in order for the payload to follow this path. The main objective of this contribution is to provide a CLPP alternative that contributes to solving the problem of rectilinear crane lifting paths. Indeed, rather than using standard robot path planning algorithms and then adapting (to some extent) the resulting trajectory to the specifics of a crane, the proposed algorithm seeks a trajectory that can be expressed as a sequence of simple rotations and translations in which smoothness— i.e., the number of rotation-to-translation transitions—is minimized. This approach is fundamentally different from previous work. In the section which follows, this paper presents the two basic building blocks underlying the algorithms described in this work: (i) the decomposition of a crane lifting path in terms of the basic motions of a crane; and (ii) a brief review on developing and merging polygonal envelopes around obstacles. In Section 3, we review some fundamental
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properties of the A* algorithm, which has been selected for this research because of its simplicity. Issues related to the topology of the paths obtained using robot path planning procedures are also discussed in the context of CLPP. In Section 3.1 an initial (and less efficient) algorithm using rotations and translations is proposed. It starts by building around obstacle envelopes shaped in the form of circular sections whose centers are the crane locations. Once overlapping areas are merged, a path following the contours of these circular sections is constructed by means of a binary tree. Although useful on sites with little congestion, this algorithm may fail to find a lifting path when the congestion increases. This is due to the fact that merging the circular sections, which often are much larger than the obstacles, tends to unnecessarily exclude large portions of the construction site. The second algorithm, described in Section 3.2, is developed to solve this very issue. In the context of this CLPP procedure, the envelopes around the obstacles are constructed in such a way as to exclude the least area of the construction site. A circular grid centered on the crane location is overlaid on the construction site and then mapped onto a matrix where cells containing obstacles receive an infinite weight. This matrix is used for path searching using the A* algorithm. Finally, we provide a case study in which this algorithm was used to assess its practical usefulness.
2. Methodology Over the past few decades, modular construction has emerged as the paradigm of choice for heavy industrial construction, allowing practitioners to reduce the uncertainty related to harsh and unpredictable weather conditions. Module fabrication, which relocates construction activities to controlled environments (i.e., fabrication plants), has become increasingly efficient as optimization techniques such as lean manufacturing are utilized. As expertise in modularization has improved, construction professionals have sought to increasingly integrate modular construction technologies into their projects. Meanwhile, the fabricated modules have become heavier and more voluminous, thus requiring high capacity cranes for on-site assembly. Consequently, minute planning of crane lifts has become a crucial activity for managers in order to ensure safe, timely, and economical project delivery. This is particularly important on congested sites where the risk of collisions between the payload and the surrounding obstacles is high. In essence, for a given load which is to be moved, at a fixed elevation, from its Pick Point (PP) to its Set Point (SP), the optimal path (in the absence of obstacles) can be described by Fig. 1, in which the object's trajectory is represented by a dotted line. Under the fixed elevation assumption, the path of any object can be represented by a sequence of rotations and inward or outward translation. For instance, the path ℘(PP → SP) from the pick point PP to the
SP
PP
Fig. 1. Crane lifting path for a load picked at PP and set at SP in the absence of obstacles. The path is represented as a dotted line.
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(a)
(b) 1
Obsstaclle
Obbstaccle
2
3
Fig. 2. Modifying a simple trajectory to avoid an obstacle: (a) collision detection and (b) collision-free path.
set point SP shown in Fig. 1 can be formalized according to Eq. (1), ℘ðPP→SPÞ ¼ ℜ ðR1 ; α Þ⊕ℑ ðR2 −R1 Þ
ð1Þ
In which ℜ represents the rotation of angle α at a constant radius R1 (with respect to the position of the crane), whereas ℑ denotes the outward translation whose vector magnitude is defined as R2 − R1. The outward translation is characterized by R1 b R2 and the inward translation by R1 N R2. As for the symbol ⊕, it is used to denote the composition of the translation ℑ(R2 − R1) and the rotation ℜ(R1, α). In practice, for any lift performed at a fixed elevation, two possibilities may be encountered: (i) No obstacle interferes with the object in which case the path of the lifted object is straightforward, see Fig. 1; (ii) One or more obstacles lie on the path of the object (or the boom). In such a case, a collision resolution procedure must be applied in order to obtain a collision-free path. In what follows, we describe an algorithm which modifies the simple payload trajectory shown in Fig. 1 in order to avoid obstacles while satisfying user-defined clearance requirements. This aspect is illustrated in Fig. 2(b) where it can be seen that even though the path becomes more complex than that in Fig. 1, its formal representation is still expressed in terms of the crane basic degrees of motions (at a fixed elevation), i.e., rotations and inward/ outward translations. According to (Fig. 2), avoiding an obstacle at a given elevation can be viewed as a process of modifying a simple trajectory, see Fig. 2a, so to obtain an obstacle-free path, shown in Fig. 2b, which can be formalized satisfying Eq. (2), ℘ðPP→SPÞ ¼ ℜ ðR1 ; α 1 Þ⊕ℑ ðR2 −R1 Þ⊕ℜðR2 ; α 2 Þ⊕ℑ ðR3 −R2 Þ⊕ℜðR3 ; α 3 Þ ð2Þ
(a)
From a high level conceptual point of view, the two algorithms proposed in this work for CLPP proceed according to the following steps: (i) around each obstacle, build an envelope representing an exclusion area which, for safety reasons, should not be entered by the payload to ensure sufficient clearance between the object and surrounding obstacles; (ii) In the case where obstacles' envelopes overlap, merge these envelopes into a single exclusion area; (iii) apply a path searching algorithm, starting at the pick point and ending at the setpoint, that builds an obstacle-free path (optimal with respect to a predefined metric). Here it is worth noting that the proposed algorithms differ in the way they handle step (iii).
2.1. Obstacle envelopes According to the high level description provided in the previous section, the first step towards determining the trajectory of a lifted object consists of constructing an envelope around each obstacle which exists within the crane's operation area. For safety reasons, any lifted object must remain outside such envelopes during its travel from the pick point to its final resting set point. The algorithm allowing an envelope to be constructed around a polygonal obstacle proceeds in three steps: (i) an initialization phase in which the first edge of the envelope is obtained, followed by (ii) a propagation step which generates the remaining edges and finally (iii) a merging phase in which overlapping envelopes are merged into a single area. The first two steps are illustrated in Fig. 3. Although the initialization of the algorithm is rather straightforward, the following implementation details are provided to enhance the readability of this contribution:
(b)
Fig. 3. Obstacle envelope generation. (a) Initialization and (b) propagation.
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Valid envelop segments Invalid segments
Fig. 4. Rules for the construction of edges of the envelope in the propagation phase.
1. Find two consecutive points A (xA, yA) and B (xB, yB) such that yA and yB are, respectively, the smallest and second smallest ordinates, that is yA = min({yk}k = 1,2,⋯,n) andyB = min({yk}k = 1,2,⋯,n − {yA}). 2. The first edge of the envelope (shown in Fig. 3) is a segment [A ' B '] such that: 8 0 0 < A B ==½AB 0 0 : d A B ; ½AB ¼ User specified clearance yA0 byA and yB0 byB
ð2Þ
In which the notation d([A ' B '], [AB]) represents the distance between segments [A ' B '] and [AB]. As for the propagation phase, the application of the distance requirement (given in Eq. (2)) yields for each subsequent edge two segments of which only one is valid since it belongs to the envelope, see Fig. 4. The selection algorithm rules are described in Fig. 4. According to Fig. 4, the valid segment is chosen according to the following rule: a. For α b π/2, select the segment for which d1 N d2 and b. for α N π/2, select the segment for which d1 b d2. Where d1 and d2 are the distances from the end point of the current (valid) edge of the envelope to the closest end points of the next edges, see Fig. 4. At this stage, the obstacle envelope is represented by a sequence of unconnected edges which needs to be connected in order to obtain a closed polygonal shape as shown in Fig. 5. For simplicity, the connections are obtained by finding the intersection of consecutive edges. It is worth mentioning that the obtained envelope is continuous but is clearly not smooth (not differentiable) at the points of connection. For the purpose of this contribution, this does not create any limitation on the subsequent algorithms but if needed smoothness can be achieved by connecting the segments by ad-hoc curvilinear arcs.
need to adapt to changing layouts, e.g., bridges becoming unusable. Because the A* algorithm combines efficiency and simplicity [8] it is among the most widely used path finding procedures in the gaming industry. The A* was built on Dijkstra's graph traversal approach but departs from its predecessor since the search from the current node (n) onward is directed by means of a cost function calculated satisfying Eq. (3), f ðnÞ ¼ g ðnÞ þ hðnÞ
ð3Þ
In which g(n) is the total cost required to get from the starting point to the current location and h(n) is a heuristic function used to estimate the cost from the current node to the final location, i.e., the target. For instance, given a layout, a starting node and a target, see Fig. 7, the A* algorithms returns the path represented in red. However, in the context of CLPP, even though the A* family of algorithms provides valid paths, their geometry does not reflect the motions allowed for a crane which as mentioned in the previous section has only two degrees of freedom at a fixed elevation: rotation and translation. As a result, for CLPPs the A* (or equivalent algorithms) results can mostly be characterized as proofs of existence of paths since their rectilinear shape does not necessarily describe what happens on the field. To illustrate this issue, let us assume that a payload is moved on a construction site free of obstacles. In this case, the application of any path planning algorithm would return a straight line which conveys
2.2. Merging individual envelopes As the construction process advances, objects placed in their final positions may interfere with the trajectories of those waiting to be lifted. As a result, envelopes need to be re-generated after each lift, and since some of these may overlap, it may be necessary to merge them into a single larger and more complicated envelope. For the purpose of this contribution, the Weiler–Atherton algorithm [11,1] was used. The result of this procedure is illustrated in Fig. 6. 3. Path searching Path searching is a well-known problem in artificial intelligence, which over the past few decades has led to the development of several algorithms [7,6] especially in connection with the emergence of computer games. Indeed, the increasing complexity of modern (strategic) games required sophisticated path finding procedures which not only need to be fast in order to ensure user-friendliness but most importantly
Fig. 5. Obstacle clearance envelope obtained by connecting the edges of the propagation phase described in Figs. 3 and 4.
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Fig. 6. Illustration of the Weiler–Atherton polygon union algorithm.
the idea of a shortest distance; however, to mimic this trajectory the crane operator would need a great deal of control on the variation of the radius of the payload. Indeed, if we assume the crane base is located at the origin (xc = 0, yc = 0), and the linear path is described by an equation of the form y = mx + b, see Fig. 8, the radius of the payload, R(θ), which varies as a function of the rotation angle (for the load to follow the line) is given by Eq. (4), 8 > < θPP ¼ b with RðθÞ ¼ > sinðθÞ−m cosðθÞ : θSP ¼
π 2 π −1 tan ðySP =xSP Þ þ ½1− sgnðxSP Þ 2 −1
tan
ðyPP =xPP Þ þ ½1− sgnðxPP Þ
ð4Þ In which θPP ≤ θ ≤ θSP is the rotation angle of the jib (or boom), (x PP, y PP) and (xSP, ySP) are respectively the coordinates of the pick point (PP) and the set point (SP) whereas the function sgn(x) represents the sign function which returns 1 if x ≥ 0 and −1 otherwise. In Eq. (4), because the payload trajectory is defined analytically, we are able to describe explicitly the coupling between the radius (distance of the load from the crane base) and the rotation angle. Even though it is challenging for (human) crane operators to precisely balance the speeds of rotation and translation which would allow the load to follow complicated trajectories, analytical forms like that in Eq. (4) may be handy for robot driven cranes. However, in the case where the trajectory is obtained by means of ad-hoc computational procedures the coupling between
Fig. 7. Finding a path connecting a source and a destination using the A* algorithm.
the radius and the rotation angle can only be done by discretization since the trajectory is not defined explicitly. In this case, the crane operator may decide to perform a sequence of rotations–translations as shown in Fig. 9 (in the case of a linear lifting path) where three different scenarios are proposed based on the length of the corresponding path. This approach which can be viewed as a discretization of the continuous motion described in Eq. (4) is time consuming since it requires many rotation-to-translation transitions. Consequently, the intrinsic degrees of freedom of the crane need to be integrated as constraints within the procedures used for payload path generation in order to avoid un-natural scenarios. In view of the difficulty in using linear paths as determined by traditional path planning algorithms, it is instructive to explore the challenges related to determining a path which uses only the basic crane motions. At this juncture, it is important to note that even though this procedure will not necessarily lead to the shortest path, it is practically more realistic. In fact, rather than determining a path and force the crane to follow it, we propose to determine a path that is adapted to the crane's constrained motion, rotation, and translation. 3.1. First algorithm: motion along the outskirt of (merged) obstacle envelops The starting point of the procedure developed in this contribution is a simple trajectory which consists of a single rotation and an inward or outward translation connecting the pick to the set point for a given object, Fig. 1. This trajectory is referred to as the Crane Simple Trajectory
Fig. 8. Moving a payload along a linear path by adjusting in a continuous way its radius as a function of θ.
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Fig. 9. Sequence of crane rotations/translations to approximate a linear path obtained by path planning algorithms in the absence of obstacles. The rotation/translation paths are ranked based on their lengths.
(CST). In the case where an obstacle interferes with the CST, an inward or outward translation is applied to the CST to avoid the obstacles (assuming a fixed elevation). When the lifted object is translated, the inward/outward movement must be within the minimum and maximum radii allowed for that weight, see Fig. 2. From an algorithmic point of view, the search for a path connecting the pick point of an object to its final resting position relies on a binary tree the nodes of with nodes are created every time an obstacle is encountered. The branches of each node allow two options: inward or outward translation immediately followed by a rotation towards the set point. In essence this algorithm moves the object along the outskirts of the obstacles lying on its path which are identified by circular sections, as shown in Fig. 10. Overlapping sections will be merged to produce a single envelope. To allow the practitioner to select an optimal path from those generated by the new algorithm, each rotation inward/outward translation is assumed to carry a penalty which is likely to be project dependent. For instance, 1. A transition from a translation to a rotation or vice versa has an intrinsic penalty which quantifies the unproductive time during which the crane operator transitions from one motion to another. 2. An object which is translated outwardly to avoid an obstacle leads to an increase in the moment due to its weight, which in turn, increases the likelihood of crane failure due to a negative reaction exerted on its base. This increased risk is particularly true in the presence of additional forces such as wind.
+
Building on the previous step, in which each obstacle was surrounded by a sector, see Fig. 10, the most straightforward algorithm allowing an object to be moved by a crane from its pick-point to its final destination consists in following the contours of the obstacles' envelopes. Incidentally, such envelopes are built using arcs and forward/backward translation, hence taking into account the specifics of the crane motions. The possible paths are generated using a binary tree to which penalties (or weights) are assigned which allows the most optimal to be selected. This concept is illustrated in Fig. 11, in which the possible paths are shown. Although the paths determined in Fig. 11 are likely not optimal since they contain several rotation-to-translation transitions, these were generated as in such a way as to be expressed in terms of the fundamental motions of a crane: rotations and translations both of which are uncoupled since each motion occurs while the other is constant. Of course, since every transition from rotation to translation (and vice versa) causes delay, the above described algorithm returns the path which contains the least amount of such transitions. 3.2. Second algorithm: motion along the outskirt of individual (unmerged) obstacle envelopes While the above described algorithm has the advantage of simplicity, it has a major drawback: merging envelopes may dramatically decrease the area in which an object can be moved hence increasing (falsely) the likelihood of impossible lifts. In the following section, we
+
Fig. 10. Circular sections around obstacles. The object is moving along segments of the envelopes which are shown in red. The crane location is identified by the white cross.
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Fig. 11. Pick point to final destination paths for an object moving along the outskirt of the obstacles. When superimposed, the individual paths on the left give the red contour on the right.
describe a more general, yet more complex, approach allowing one to obtain an optimal path for a given object. This algorithm proceeds in four steps: 1. Apply a rotation and a translation to the site so as to put the pick location on the X axis and the crane location at the origin. 2. From the location of the crane (xc = 0, yc = 0), draw rays connecting the crane to each vertex of the obstacles. 3. For each vertex (including the pick and set points), a circular arc is drawn starting from the X axis (containing the pick point) to the ray which contains the set position. 4. The intersections of the rays with the arcs are compiled into a matrix, in which the columns correspond to the angles (smallest to largest) and the lines represent the radiuses (smallest to largest). Nodes which fall within an obstacle are given an infinite weight. Given the matrix (which can also be viewed as a graph) of the possible nodes, the next step consists of applying any known algorithm (including the A* algorithm) to find the shortest path between the node representing the pick point to the final destination. The matrix corresponding to the case depicted in Fig. 11, which was made more complicated by the addition of two circles representing the minimum and maximum reaches of the crane, is given in Fig. 12. In this table, “x”
Fig. 12. Matrix representation of the ray–arc intersections. Red symbols denote forbidden locations. Blue indicates the pick and set points of the object.
denotes the nodes that fall in forbidden areas (infinite weight), whereas as “1” indicates allowed positions. Although the matrix in Fig. 12 (which essentially is a graph) provides a number of possible routes linking the pick to the set point, in practice one may choose the path with the least number of rotations to inward/ outward translation transitions. An example of such paths is depicted in Fig. 13 in which the most optimal route contains three rotation-totranslation transitions. Note that in the layout described in Figs. 12 and 13, the payload was purposely forced to be moved in the upper half of the site, i.e., the most congested area, to illustrate the flexibility of the procedure described in this work. In practice, however, the algorithm described in this paper above allows the boom to rotate in both directions: clockwise (lower half) and counter clockwise (upper half) and returns the smoothest path.
4. Case study The algorithm developed in this work was applied to a case study in the field of heavy industrial construction site where heavy modules needed to be assembled into a complex structure to be used in the oil industry in Northern Alberta. In this context, a typical project may require assembling 50 to more than 300 modules. As the project progresses, the construction site becomes increasingly congested
Fig. 13. Selected paths constructed from the matrix mapping of the ray–arc intersections. The inner and outermost circles represent the minimum and maximum reaches of the crane.
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SP
PP Fig. 14. A plane and isometric view of a module to be lifted by a crane from a Pick Point (PP) and installed at the end of a corridor like structure identified as the Set Point (SP).
hence requiring minute planning of the remaining lifts in order to ensure a level of productivity allowing scheduling and safety constraints to be met. This is particularly important for the Northern part of Alberta where the harsh winter conditions (temperature and wind factor) can be challenging. To illustrate the usefulness of the algorithm developed in this work, we provide a case where a module picked from a location (PP) is required to be installed at the end of a corridor-like structure at a location SP, see Fig. 14. In the case portrayed in Fig. 14, a DEMAG CC2800 crawler crane whose maximum capacity is 660 t at a maximum boom length of approximately 84 m (275 ft) was used. The crane was equipped with a superstructure to ensure that all modules could be assembled safely. The module of interest shown in Fig. 14 weighed approximately 65 metric tons for a size of 12 × 6 × 7.21 m. Referring to Fig. 14, the module of interest, picked from a trailer, Fig. 14(b), is required to be installed at the other end of a two-part structure which exhibits a corridor like area, Fig. 14(a). Although, the module could be lifted above existing structures, the preference of our industrial partner is to lift as close to the ground as possible before considering lifting at higher elevations since the former scenario provides a better control of the load especially in the presence of strong wind. Given the pick and set positions of the module shown in Fig. 14, the application of A* and the C-space algorithms returned a linear path which requires the load to be moved in the corridor since both of these procedures target the shortest distance as their objective function, Fig. 15. However, although valid, these paths were found not convenient from a practical point of view since they require a great deal of control in order to move the module of interest without collisions with the surrounding obstacles. Conversely, the procedure developed in this work was able to determine an alternative path since the metric
underlying the validity of a path is its smoothness measured in terms of the number of transitions from rotation to translation (and vice versa). As a result, given the fact that the reach of the crane used for this operation allowed for the module of interest to be moved at a larger radius, the path from the pick to the set point in the exact same form as Eq. (1), that is one outward translation, one counter clockwise rotation followed by an inward translation. This type of paths which are constrained to use the basic motions of a crane turned out to be easier to follow by crane operators. 5. Conclusion Cranes are among the most important equipment used in today's construction industry, especially in the context of modular construction, where heavy modules built off-site need to be assembled on site according to time-constrained schedules. This is particularly true in the case of Northern Alberta which is characterized by harsh winter conditions which can reduce the window for outdoor work. As a result, cranes, which over the years have increased in size and capacity to accommodate the requirements of modular technology, have become a critical resource on construction sites. The utilization of these cranes, though, requires minute planning in order to reduce the risks of failures and delays. In this respect, optimizing this resource has become an important task that can no longer be overlooked by project planners and practitioners. The main thrust of this contribution is thus the development of an alternative CLPP approach which has the advantage of generating paths that are constrained to be represented in terms of the basic motions of cranes, namely, rotations and translations. In fact, the main issue in using traditional path planning algorithms is the shape of the resulting paths, which are expressed as a polyline. From a
Fig. 15. Module path as obtained by the A* and C-space algorithms which optimize the shortest traveling distance.
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mathematical point of view, this topology which does not take into account the specifics of the crane motions ends up coupling the radius (distance of the payload from the crane) to the rotation angle in order to mimic trajectories which are not easily expressible in terms of the intrinsic motions of a crane. In practice this could be carried out by precisely controlling and balancing the rates of rotation and translation which can be very challenging especially for inexperienced crane operators working in congested sites. As a result, rather than applying path planning techniques that ultimately lead to linear paths, this work proposes to explore the possibility of generating paths that can easily be expressed as a sequence of rotations and translations in accordance with the basic motions that can be performed by a crane jib (or boom). The proposed algorithm aims at complementing existing CLPP procedures in order to improve the efficiency of crane operations. Acknowledgments The research presented in this paper has been financially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), project number CRDPJ 380518-08. References [1] P. Atherton, K. Weiler, D. Greenberg, Polygon shadow generation, Comput. Graph. 12 (3) (1978) 275–281.
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